Research on the historical development of analytical mechanics that was applicable to the old quantum theory

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This research examines the historical process that the so-called Hamilton-Jacobi theory, whose proto-type was established in the middle of nineteenth century, became applicable to the old quantum theory. Among the various aspects of this theory we highlight is the notion of the canonical transformation. The canonical transformation was one of the most important essential factors for the application of the theory to the old quantum theory.We note that the Hamilton-Jacobi theory was effectively used not in physics but in celestial mechanics in the nineteenth century. By examining some articles of the celestial mechanics published just before the birth of the old quantum theory, we find the following facts: It was Poincare that first mentioned the relation between the canonical transformation and the complete solution of the Hamilton-Jacobi theory in his large work of Mecanique Celeste published in 1892-99.Charlier gave an exact proof of this relation in his Mechanik des Himmel in 1902 and 1907. He in fact transformed the canonical equations of motion and showed how the angular variables, which often used in celestial dynamics, become the canonical elements. Schwarzschild referred to this result and succeeded in constructing the action and angular momentum that provided appropriate mathematical tools for explaining the Stark effects. Schwarzschild's new canonical variables effectively used in explaining the old quantum theory were essentially derived from the results of celestial mechanics.